# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068196:0398
# Permalink to this reading: https://leibnizlegible.com/page/00068196:0398?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 201v, page id 00068196:0398
# Scan: The same scan is also registered as fol. 202r; its text is given once, here.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=398
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000398.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 185 lines, status machine
# Lines: 185 recognised, mean confidence 0.77
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# After the first blank line: the recognised lines in reading order, one per line.

CL A55I5 FRIMA
alerilus 1 10 X + 1 2o 2oX + 2a = 0 sp. 2ox u nA = (o2oB + 10ua fit 10.  - 11 20 = 0. elter 1B 217  11 21) = 1 20XF 11 21a) fiet idem
lalus son 10X L Ha =0, 2OHCFINXF aA Do (IB 2OXXSH I. EOAX = (B. 20KDHIB LI AXHIB L2aa fit , 1B. H aX + 102 aa 20 quae conjungatur aepatim 1. frot 10. 10. 32 = 10. v. 21 4 111. 2020
eq. A si pro 10 gubstitua permutet 0 et
Aliter 1B22A XFH. ma) = 11. 2oNT + 11. Uo VanD (., ve 12oi =  F quaesi compimgatur 5p
q lum havetimar te di ddiu
puitaeq. 4. Hinc duplex processus her irit
at esqu. llmnam callauli (m posito 22
ciluus 1eXtIa = , 20XPXt 2NX + 23a=0 (10. 20X + 11. 20 axx = (10. 20V Ho. 2axY F 1v. VaaX + 10, 23 a  seut 10. 21 ayY + 10. raaX +. 1B. 13 a La consendi 2op tyax   aa 
putiontum 
un l nlepire s
ncauli prrecedentis, si ponis ho-0 et pro  l p2, 29lice, nonas 2o, X1, 22. Eotem modo ex aeque ultina pasculi pre  b.  
3) calculi
pentinaequ. d
Aequationes quartae calculi (2, 3, et. consentiunt inter se in eo quod ex Ata calculi alicujus fit quarta calculipraxim e a u sipliando ilam per X, et adento   10. 2C a. Et ita paht etiam quomodo quintae ex se invicem criantu, et ita pur
10. 10. 11. 22 + 10, 11. 11. 21 = I1. 11.
10X . H
sequationes 3tiae  calculi 1. Ed, 3ti, consentiunt inter se in eo quod ex Htia Ealuli aliujus,  sit otia calculi prorime sitelianto vtamm per X, et adendo 8  B
poncido (20), (21) (22) calcili hanus, et fit
remotae a primus nempe 3tiae, quantae, quintae _c. a se invicem ducantur Vibeamus et quomodo a se invicem ducantur Uttimit
suamncaliani
Quaeramus praeterea residuns.
tet berdiaa
s
ex au
10X LIIa
divitor 10X + 11a
noru ntrcdae rattis 2tit  ua
e Resid. )- poero
penmetimae, et antepenultima
comite decurem lessum t hae mat
et Residuus Cum
tione altina salub
pisisur 10X + 1
uinistox t LlaE + DOaX Y2
e - meaet -e ut 
Caeterum aliter et brevius poterit hic tolli litera X, si ex aequ. 1 fial X = - 118 et i valor
unc aequationem conjongamus cum sequentibus
I0XX F 1af + 1244 = 0 2OXX + 2IaXF 22aa = 0 (10207X + 11. 20aX + 12. 10aa = (10. 20Xx) + 20 21 ax + 10. 22a4, fit. F10. 21 ax
sibstituatur deinde in aequatione secarda
CLASSIS SECVNON
Sed tunc transeamus ad aequationem 10 XX +  I1aX + 12 aa,
alculys primus
tit ex lalentosecundo clastiis praecedentus sidio 22 (oi) 2t 1 (t 1) 20 = 0 _ ubi explicando 1ou lt, habelatur equatio fine p.
Unde ex eae C et 2 fet
 10. 22- 12. 20  ( 1021 41. 20d a X ad xarsus = fi. 2l 2
in hoc calculandi modo id est laudabile, quod hoc modo non a
Lode
calculus secundus 10xX. + H1aX +1 2a4 =. 20X + 2XX + 22 aax + 23 a =0
unde Ex his priorem applicemus primae datae inversae, fiet 18. 16. 23 p
rrunt depressiones
x)Hisorvaret (2. 20 aaX = (10. 20X1
ax, 
et pradbit (18. 12. 20XX L 11, 2. 20 aX + 1212. 20 aa = (0. 12. 2oXY) + 0. 12. 21 ax + 10. 12. 22 aa, sed aequatio quae ita prodit coincili
Rarsus et posteriorem Minvensione _ (1) ex inversioneortam appliaemus primae datiei
 quod si ex aegu. Micte data, et rda inventas ada modum
Calculi rdi
nec 2dae ditae ntra gradum 2dum Ego aequationem pristem datam, et ap. ) conjuaga
+ 10. 10. 22 X +10. 10. 23a 20
- 10. 12. 20
-10. 12. 21
+ 11. 12. 20
similite raeqrationem priorem datam, et aequo 2) conjana
10. 12, 22 X +
seu = 10. 1
= 2. 12. 20
I. conjungere ejus inverso
1o t H.2 B
(10 12. 22. 23) 10. 1 . 2 =10. 12. 21. 22), 10. 1. 2. 2 (C 10. 12. 16 23  10. 1. 1. 20. 23 + 10. 123. 20. 2 PoV. 15. 20. 22 -16. 11. 12. 21 23 +18. 1. 2t 3 410 4. 1. 4 - 1e se M 21. 2e to site ro quiao 8. listi ton e . 3
agn. 10. .25 (+18. 12 22. 19 (1 5 12. 24. 13) + 10. 11. 12. 21. 23(-15, 12. 21. 22f 10. 13, 2824 410. 11 12. 23 + 10. 11. 12, 2822 =11. 123.20. Qu.aqvatione defantus destnctis destruendis duisa per i prodit
FI. l. 20
18 20
+8. 12. 25
. o
Geerali via, ien usi dita aequatont 104 XBE, = B e ron A 2B AuX L2EXL = 13 7d +2A X +etu pestumus touco X, soum in aequ. posteiore pro X poreraer - 1B. unde fiet a
= 20 - 19  + 1gue n - 1 13 4 1 24 -  1825 + et e vel  18. 20 - 10. 18 al + 1. . 22 + 18. 10 3 23 - 11 10 24 + tc aente periveniatur au  plae qui ex terminus altimus.
se sint jum intre agus. D. + IX + 1BXY - 0. et 20F LIY F LEAX +23 X + 14 X 425 X + 36 X + et = 0. Eit XX    = io  adege N= 2 1 + sia1
quo valome substituto in aqu. aleradata fist 20 B. v + V2- N 23+ 1 2 1 - 1 25 + erc 
20C - 1, 21 +   22. sl 23 + 1  1 25 + re s l 17 ten = sa2. 21 + _2 B + 42 95 + 82 27. Quod si san detimus tollere irrationalem, fiet
20 -  11. 20. 21 + 2, 1 + 42. 20. 22 - 2. 18 20. 23 + 2, 1 4+ 4
x
16. 104 8  2 11 2025
+ n 2121 - 25 .F2 212 + 2
Unde t deptrum latus aequationis in sinistrum transfereudo, fiet:
+2 1ror 286 - et
+ qeate vu - 2. 1 + dta 23 + 2, 1E 3  v 24 - et
20- B: G8 1. 2021 + 2 ( 19 62. - 2 (L. 13 20. 1.
(24 3 2
81 ) L. p8 Zu2e
A sertet sujus lex habetur ex conideratione numerorum 20, 21, 22, 23, etc. quea quorum
hic ponuntur  quadrata, et biniones. e quoties hujus numeri inota dextra est
impar, ut cum est 2t, 23, 25, ex. pefficiens habet factorem H, H3, 1, etc. Cum
Quoties nomerus cujus neta dextra est par, ducitur in cum nume
64. 196 23.23 - etc
+ 2 511t 12024
1610 e 3 223
Numerus  cujus nota dextra est  impar, ut 2, 23, 25, habet in eoefficiente 11, 11, 11, etc
par ut re. 24,26, ex habet in eofficiente t, 112 412, 114, etc
Hine si combinetar par, v. g. 22. 26, coefficiens est productum ex 11+ 42, in 115+ 42
Si combinetur pari par m et impar, V.d. 22, 25, prodnctum coefficiens est productum ex 114+ 42, et 155
eisere
denique si combinetur impari, productum est coefficies est Vg. 23. 25, paoductum coesficiem
quidem deberet esse iP in 15, sed to seu 118, sed tamen ab eo detrahitur A2, quod homogeneum illier
ox 18
(A 121. 25
 Fi pu 2+
+H 277
n1 Y21. 27
 fr. 26
+111, 723. 25
(1.   3 4 28
rt H3, uti revera assurgitur uetra 123
videndrem ergo in applicationequ generalis (anonis ao speciem temper justo altroea destruantur
iscubica. ita seme cubando et seut quid anto aequaturera ratomlem liberam . X logtqutum
atexemaltimae ete. Nempe allima calculi propo alicujus dabo
nea et ut suceit divieo fit l0 usante
unde per
 utto ex e aeqam alisuno eru
F1ClI. 23aae)
Coincidet Residu  issue a
potens sero compo-
netur ex on
equationtl ultimatur
calculorum praecedentium
tiplicario fiss)
primo sollere velimus nimis alt
hultie fers
io. 12 ux t un ax ( H
t o ita eleganter evenit, ut dum in und
et fiil +10. 10. 22
in 20X + 2IaXX + X+ 25a, item cum 20X + 2CX +. 22aCL +25a X 24a
eaa stio
XX) F 10. 11. 21aX
pt avierus n
rexFua
easse  HeumX
 12. 20a
eptinum
papaer
 ut t opu 2. iacentus
tionis prim
42) + 10102E ax + 10
oru at
2. 2 = 12 12 204 121. 21 ax ( 1
 vel atiqui termini destut
= + 10. 10. 237f- 10. 11, 237
2d a 20
simplex in altero 12 assurpit ad
nota ulaeti palicata
divitentur per
 1220 redit
 raa 2.
modum classs
seu fiet
Ehratum et co est sumples
sed quia it utoblu alagit ad p    a  b
3 savit ad quadratum et iL es
habentar nili 103 et
=0. se
10. 22. 2C
destuentat aeliqua
abest H, et renqua
a  et fiec
si esset 11 20 fiuet  Et proinde
X = 126 = 10. 222
- 20. 12 ac + 21 ax + 2 -  0 seu
XX = qu 18.2 p rgaa = - seu 1  1 o. 2. 20
1o  aa o exsecta eia sndet quanmenticnem
10. 22 aax + 1B. 23a. fit 10
1923 AX( 10.12. 2344) = +. 10. 12. 21 XX + 10. 12. 12 ax (4 10. 12. 23aa) Unde podit  ade
mcedenti
uddettis proe
2. 10. 12. 20. 22, hic autem est solum 10. (2. c8. 2
42, 21 + 2. 42 ∞23+ 2. 82 2125 +2. 41 227 1 2. 47, 2129 + ex
vs +  3as
+2. 48 2327 +eu
155 8 2525. + etc
(V1. 20. 25
3210) 15
1.425 E1. 24
18.61) 11. 20. 27. Fib . 8. 1a
ut pu. 26
1 + ) 3.24
Seb videtue hic altius assurgi quam oporiat, ex. gr. cum non assuru im ultra 23. videtur
perveniri ad 116, et 106, cum lamen non debeat assurgi intra